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Pythagorean Theorem Algebra Proof

What is the Pythagorean Theorem?

We have a page that talks all about the Pythagorean Theorem, but here is a quick summary:

The Pythagorean Theorem states that, in a right triangle, the square of a (a²) plus the square of b (b²) is equal to the square of c (c²):

a2 + b2 = c2


Proof of the Pythagorean Theorem using Algebra

We can show that a2 + b2 = c2 using Algebra

Take a look at this diagram ... it has that "abc" triangle in it (four of them actually):

Squares and Triangles

It is a big square, with each side having a length of a+b, so the total area is:

A = (a+b)(a+b)

Now let's add up the areas of all the smaller pieces:

First, the smaller (tilted) square has an area of   A = c²
And there are four triangles, each one has an area of   A =½ab
So all four of them combined is   A = 4(½ab) = 2ab
So, adding up the tilted square and the 4 triangles gives:   A = c²+2ab

The area of the large square is equal to the area of the tilted square and the 4 triangles. This can be written as:

(a+b)(a+b) = c²+2ab

NOW, let us rearrange this to see if we can get the pythagoras theorem:

Start with:   (a+b)(a+b) = c²+2ab
Expand (a+b)(a+b):   a²+2ab+b² = c²+2ab
Subtract "2ab" from both sides:   a²+b² = c²

Now we can see why the Pythagorean Theorem works, or, in other words, we can see proof of the Pythagorean Theorem.

There are many more proofs of the Pythagorean theorem, but this one works nicely!

Note: parts of this page are courtesy of WikiBooks